Mixed convective flow of a magnetohydrodynamic Casson fluid through a permeable stretching sheet with first-order chemical reaction

This research article presents the magnetohydrodynamic Casson fluid flow through an extending surface embedded in a porous medium. Furthermore, the Casson fluid flow is investigated under the effects of thermal radiation, Joule heating, viscous dissipation, and chemical reaction. The analytical solution of the modeled problem is utilized with the help of homotopy analysis method (HAM). The convergence region of the applied technique is portrayed graphically. The impacts of the embedded factors on the flow profiles are exhibited with the help of figures. Furthermore, numerical values of the surface drag force, heat, and mass transfer rates are highlighted via table. The results show that the augmented Darcy number, Casson and magnetic parameters have declined the velocity profile of the Casson fluid flow. Growth in Brownian motion augments the chaotic motion amongst the particles due to which the kinetic energy of the particles transforms to heat energy which consequently augmented the thermal profile, while reduced the concentration profile. The mass and energy profiles are positively effects with the increment of thermophoresis term. And the growing values of chemical reaction and Lewis number cause a reduction in the diffusivity of mass of fluid due to which less transfer of mass takes place that weakens the concentration layer thickness and declines the concentration profiles.


Introduction
During the past few decades, the mass and heat transmission past a permeable and extending sheet has achieved a significant response from numerous researchers for its important applications in industry and engineering technology. Carbonell and Whitaker [1] inspected the transmission of heat and mass for permeable medium with the main focus on the energy and concentration equations in their study. Yaglom and Kader [2] have investigated the transportation of mass and heat within a turbulent fluid flow and a rough wall by employing a high Reynolds number. It has been observed in this work that, more disturbances have been produced due to the roughness of the wall that has augmented the transmission of heat and mass in comparison to a smooth wall at similar values of Prandtl and Reynolds numbers. Bandaru et al. [3] have scrutinized the thermal and mass flow from a rotary cone to fluid transport through a permeable medium by using thermophoresis and nonlinear convective impacts. The thermal flow is more prominent and influential than mass transportation. Krishna et al. [4] have used a permeable vertical sheet to investigate the temperature and concentration for a second grade fluid. Ahmad et al. [5] observed the transportation of heat for hybrid nanoparticles flow upon a porous surface.
In fluid flows, the transportation characteristics have been described by two different models that are Buongiorno model [6] and Das, Tiwari model [7]. The main focus in the former model is upon the upsots of thermophoresis and Brownian motion regarding the flow of fluid under the influence of numerous flow conditions. Khan et al. [8][9][10][11] conducted remarkable work in the field of energy and mass transference for fluid flow employing different geometrical and several flow conditions views. It has been highlighted in these investigations that the energy field has augmented by varying the Brownian motion parameter and thermophoretic effects, which on the other hand has declined the mass flow rate. Mittal and Petal [12] have discussed the impacts of these two terms over mixed convection bi-dimensional MHD fluid flow by taking the heat generation and nonlinear radiations. It has been observed in this study that the concentration characteristics have declined while the thermal profile upsurge with augmented values of thermophoretic. Ashraf et al. [13] have inspected the effects of these two terms upon the flow of fluid using viscous dissipation in the closed vicinity of the stagnation point. Abdelmalek et al. [14] have used the electrically conducted visco-inelastic nanoparticles flow on a stretching surface under the influences of thermophoretic effects. It has been detected in this inspection that flow has reduced and thermal profiles have increased with augmentation in magnetic strength. Rashidi et al. [15] presented the comprehensive analysis of the thermophysical properties of hybrid nanofluids. Mahdi and Nazari [16] investigated the water-based Ag nanofluid flow by using an artificial neural network.
The combination of chemical reactions with the analysis of energy and mass allocation plays a momentous role in flow problems. For its importance and extensive applications, it attained the remarkable attention of researchers and scientists in the last few decades. These applications include different engineering progressions such as cooling of electrical equipment and devices, transpiration of humans, chemically catalytic reactors and aircraft propulsion devices, etc. Gangadhar and Bhaskar [17] have presented a two-dimensional mathematical model to discuss the thermal and mass flow for MHD fluid flow upon a porous and expanded surface. It has been revealed in this work that, the flow profile has declined sharply due to growth in the magnetic field while the thermal flow has amplified in this process. Seth et al. [18] inspected the upshot of chemical reactions upon thermal and mass transmission for convective fluid flow past a moving porous sheet. In this work, a closed form solution has been obtained for flow, energy, and concentration equations by using Laplace transformation. Moreover, the descriptions for Sherwood and Nusselt numbers have also been determined in this work. Reddy et al. [19] have presented the profiles of thermal and mass transmission for MHD fluid flow by using chemical reactions on a rotating permeable disk. The thermal radiation upshot and partial slip has also been incorporated in this work and has established that flow has reduced while thermal flow has improved with the increment of solid nanoparticles.
Raju [20] has inspected the time dependent MHD fluid flow on a penetrable and inclined sheet by considering the chemical reaction upshots in the concentration equation. In this study, the Darcian flow model for permeable surfaces has also been used in the flow system. Punith Gowda et al. [21] surveyed the fluid flow and heat communication for non-Newtonian fluid using a chemical reaction. The results of this work revealed that flow has augmented and thermal profile has declined with escalating values of Marangoni number. Moreover, it has been perceived that the flow profiles condensed with the expansion in permeability term. The readers can further study the effects and benefits of chemical reactions upon fluid flow system in Refs [22][23][24][25][26].
A surface that contains small void spaces (pores) is characterized as a permeable surface. The idea of the porous medium is extensively used in applied sciences such as filtration purposes, petroleum engineering, petroleum geology, geophysics, biophysics, and biology, etc. Due to its importance in the field of applied sciences, many investigations have been carried out with the core focal point on thermal flow and transference of mass in fluid motion upon the porous medium. Bejan and Khair [27] have discussed the heat transition for the fluid flow over a porous texture. Chaudhary and Jain [28] have investigated the fluid flow past a plate implanted in a permeable region. It has been established in this investigation that magnetic effects have supported the thermal characteristics and have opposed the flow of fluid. Jiang et al. [29] reviewed the MHD unsteady flow of fluid inside a penetrable surface with the influence of Hall current upon thermal flow. The concept of fractional derivative along with constitutive equations has been employed in this work. Kumar et al. [30] have inspected the cross flow for MHD fluid upon an accelerating upright surface in an absorbent surface using Hall current. It has been observed in this work that mass transfer of fluid has been maintained by enlarging credit of Soret number and has been opposed with augmentation in Dufour number. A comparative study has also been carried out in this investigation with a fine agreement with published results. Haq et al. [31] scrutinized the convective thermal flow of ferrofluid across a porous exterior. In this study, the thermal radiation has been incorporated in the temperature equation that has supported the flow and temperature profile.
After a careful review of the above literature, we are in a position to present the steady and incompressible two-dimensional flow of magnetohydrodynamic Casson fluid through an extending sheet inserted in an absorbent medium. The fluid flow has exposed with the impacts of Brownian motion and thermophoresis by using the idea of Buongiorno's model. The influence of chemical reaction has also been incorporated mathematically in the concentration equation. At the end of this analysis, we will respond to the subsequent research queries.

Problem formulation
Consider the steady and incompressible magnetohydrodynamic and thermally radiative flow of Casson fluid across a stretching sheet embedded in a permeable medium. The extending velocity of the sheet is defined as u w = ax where a is the positive constant. In the xy-plane, x is taken along the sheet and y is taken perpendicular to the stretching sheet as shown in Fig 1. A magnetic field of strength B 0 is applied normal to the fluid flow. The gravitational force g acts in downward direction. The surface temperature is represented by T w whereas the ambient temperature is signified by T 1 . Also, the surface and ambient concentrations are respectively represented by C w and C 1 . The leading equations are defined as [32,33]: with boundary conditions: Here, the velocity components are u and v along x-and y-directions respectively, v f is the kinematic viscosity, ρ p is the density of the nanoparticle, ρ f is the density of the fluid, D B is coefficient of Brownian diffusion, D T is the coefficient of thermophoresis diffusion, α f is the thermal diffusivity, kr is the chemical reaction term, C is the concentration, T is the temperature, C w is the surface concentration and μ f is the fluid viscosity.
The correspondence transformations are defined as [34]: Using the above resemblance substitution, the leading equations are reduced as: Here, the magnetic parameter is indicated by M ¼ the buoyancy ratio parameter, thermophoresis term is denoted by , and the porous medium parameter is represented by The mathematical expressions for the physical quantities are stated as: where τ w , q w and q m are rebound as: Eq (11) is reduced as: where is the local Rayleigh number.

HAM solution
The initial guesses and linear operators are taken as: with properties: where q 1 − q 7 are the constants of general solution.

Discussion of results
This work describes the improvement of energy and mass transmission for a mixed convection Casson fluid flow upon an extending sheet using an absorbent medium. The fluid flow has exposed the impressions of thermophoresis and Brownian motion by using the idea of Buongiorno's model. The influence of chemical reaction has also been incorporated mathematically in the concentration equation. The equations that regulated the flow problem have transformed to dimensionless notation with suitable similarity variables. The solution has been determined by HAM. The codes for HAM are incorporated in MATHEMATICA 12.0 software. The authentication of the present analysis is represented by Table 1. The effect of various substantial constraints on the fluid flow profiles has been reviewed in the subsequent lines. To support our discussion, a graphical view of different profiles has also been presented. Fig 3 portrays the influence of Darcy number Da and Casson parameter β upon the velocity profiles. It has been spotted from this figure that with enlargement in Darcy number the void spaces in the surface will increase that offers more resistance to the fluid motion. In this physical phenomenon, the velocity characteristics decline. On the other hand, the declining impact of Casson coefficient β on the velocity field is depicted. The reason behind the declining impact on velocity profile is that the augmenting Casson parameter augments the viscosity of the fluid flow which yields the reduction in velocity of the fluid flow. Also, the increasing Casson

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it can be explained as a growth in Nb augments the chaotic motion amongst the particles, due to which the kinetic energy of the particles transform to heat energy. In this process the thermal profile of the particles grow up. Similarly, a growth in thermophoresis constant Nt supports the energy field. Actually for upsurge values of Nt the difference in temperature of the fluid grows up due to which more heat will transfer from hotter to colder region. Hence augmentation in Nt increases the thermal profile for fluid flow system. Fig 6 depicts  extension in magnetic parameter leads to generation of Lorentz force that causes a reduction in the flow profile due to resistive force. In this process a maximum heat flow takes place and causes an augmentation in thermal profile. Fig 8 expresses the upshot of Lewis number Le and chemical reaction factor K upon concentration profiles. The augmenting values of Le and K causes a reduction in the diffusivity of mass of fluid, due to which less transmission of mass takes place that weakens the concentration boundary layer thickness. On the other hand, the molecular diffusivity also suppresses with the growing in chemical reaction term which eventually decays the concentration of the fluid flow. Therefore, a declining impact is depicted here. Fig 9 examines the impacts of Brownian motion Nb and thermophoresis factor Nt upon the concentration profiles. The higher values of Nb has a reverse impact on the mass transfer. Actually, with the augmentation in Nb the random motion boosts due to which fluid particles are colliding more frequently. In this phenomenon more resistance is experienced by the fluid motion that weakens the concentration boundary layer thickness that ultimately drops the mass profile as portrayed in Fig 9. The concentration gradient within the fluid particles

Conclusion
This work describes the improvement of energy and mass transmission for a mixed convection Casson fluid flow upon a stretching porous sheet. The fluid flow is considered with the impacts of thermophoresis and Brownian motion by using the idea of Buongiorno's model. The influence of chemical reactions has also been incorporated in the concentration equation. The equations that regulate the flow problem have been transmuted to dimensionless forms by using suitable similarity variables. The solution has been determined by HAM. The impact of numerous substantial constraints upon different profiles of the flow systems has been discussed in detail. The key points of the proposed analysis are listed as: • Growth in Brownian motion augments the chaotic motion amongst the particles, due to which the kinetic energy of particles transforms to heat energy which consequently augmented the thermal profile, while reduced the concentration profile.

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• Augmentation in thermophoresis parameter has increased the temperature and concentration profiles.
• The augmented radiation parameter, magnetic parameter, and Eckert number have augmented the thermal profile, while the higher Darcy number has declined the thermal profile.
• The growing values of Lewis number and chemical reaction parameter cause a reduction in the diffusivity of mass of fluid, due to which less transfer of mass takes place that weakens the concentration layer thickness and declines the concentration profiles.
• The higher stretching parameter has declined the thermal and concentration profiles.